[Math Lair] Indian Mathematics History

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There were several talented Hindu mathematicians of note during the medieval period. The Indians and the Arabs traded with each other, and so the work of Hindu mathematicians influenced the work of the Arabs, and in turn the work of the Arabs would later influence European mathematical work. Important mathematical concepts coming from India were those of a place value system, Hindu-Arabic numerals, and the idea of zero as a number. The origins of these related innovations are obscure, but they were likely developed around 200 B.C. or so, and slowly gained popularity over the next several centuries.

The earliest known Indian mathematician of note is Arya-Bhata, who was born at Patna in the Gupta Empire in the year 476. His chief work is the Aryabhatiya, which he wrote in 499 at the age of 23. Three of the book's four parts concern astronomy and trigonometry; the remaining part lists various rules of arithmetic, algebra, and trigonometry, including the sum of the first n natural numbers, squares, and cubes, and a formula for solving quadratic equations. No proof of any of the statements in the book is given, and several of the geometrical propositions in the book are inaccurate.

Brahmagupta is the next Indian mathematician of note. He is said to have been born in the year 598. His main work is the Siddhanta, in which two chapters are devoted to arithmetic, algebra, and geometry. One of the more interesting parts of the work deals with what are now known as Diophantine equations.

Unlike in the work of Arya-Bhata, there are some attempts at proof. For example, there is a proof of the Pythagorean theorem. Bhaskara also provides some geometric results, including the areas of inscribed triangles and quadrilaterals, and attempted to calculate an approximation to π; (although his best approximation is no better than 10, which is only accurate to one decimal place).

Brahmagupta may have been acquainted with the works of Arya-Bhata and Diophantus, but it is most likely that much of this work was original.

The next notable Indian mathematician was Bhaskara (1114–1185). He is said to have been the lineal successor of Brahmagupta as the head of an observatory at Ujein. The Arabs were aware of his work, so the work became indirectly known in Europe after it was written. His main work is the Siddhanta Shiromani, written around 1150, which consists of four chapters. The first, Lilavati, is on arithmetic; a second, Bija Ganita is on algebra; the third and fourth are about astronomy and the sphere.

Lilavati contains a large amount of material. The decimal system is briefly described. While Hindu-Arabic numbers and the decimal system had been used for centuries before, this is the first work that systematically describes the decimal system. Bhaskara defines eight basic arithmetical operations (addition, subtraction, multiplication, division, squaring, cubing, and taking the square or cube root). He gives various properties of zero, showing that he, like Brahmagupta, considered zero as a number. Quadratic equations and simultaneous linear equations are discussed, as are the sums of squares and cubes, problems involving interest, an approximation to π, and many other topics.

Bija Ganita uses a detailed mathematical notation, including signs for subtraction (addition is indicated by juxtaposition), products, fractions, squaring and cubing, and symbols for unknown quantities. His method of writing equations represents an advance over previous work. Polynomials are arranged in powers. Many equations have numerical coefficients, with the coefficient always written after the unknown. Positive or negative terms may both be allowed to come first, and every power is repeated on both sides of the equation, with a zero given as the coefficient if the term is absent. Bhaskara devised a method for solving various indeterminate equations, and he also found a method for finding the solutions for what is known as Pell's equation.

The other books contain various information on trigonometry, including some original trigonometric work. There are also preliminary concepts of calculus, although this preliminary work was not further developed.

Perhaps the only other pre-modern Indian mathematician of exceptional note was Madhava (ca. 1350–ca. 1425). While it is not always clear which works attributed to him were in fact his own and which were those of his successors, his work includes work in trigonometry, a value of π accurate to 11 decimal places, mathematical analysis, and further preliminary concepts of calculus.

The book History of Hindu Mathematics: A Source Book is an excellent source for learning more about pre-modern Indian mathematics. Several sections of the book can be found on Math Lair.

You may also be interested in a selection of problems from classical Indian arithmetical works, from the above book.

Sources used (see bibliography page for titles corresponding to numbers): 30, 37.